An empirical approach to the geometry of Eupalinos' tunnel on Samos[edit]

Abstract: The Eupalinos tunnel is the most complex structure that has been preserved from the ancient world. The tunnel was cut from both ends, 1040 m through the Kastro mountain on Samos, around 550 to 540 BC and it has fascinated people ever since. But still more intriguing is how Eupalinos navigated, because the tunnel zigzags in a most challenging way. By estimating the accuracy obtained by Eupalinos from the geometry of the construction, the author reconstructs the methodology of the tunnelling. He proposes that Eupalinos used the straight line as the main component for his navigation, that he set out tunnelling directions at a few precisely defined turns using simple geometry easily derived from the straight angle down to a unit not smaller than 7.5 degrees and that a tangential-based method was not used. He argues that Eupalinos used a full-scale model to determine the geometry of the tunnel. He suggests a new system for the length measurements of the tunnels based on the wall marks. He argues that the design of rendezvous is based on necessary catching width between the two tunnels. He proposes that it is a coincidence that the rendezvous takes place close to the summit. He argues that Eupalinos made gross errors in handling the measurements, but that he managed to handle these errors, probably thanks to careful remeasuring.

The Eupalinos tunnel was cut from both ends, ca. 1040 m through the Kastro mountain on Samos, around 550 to 540 BC and it has fascinated people ever since. The construction carried drinking water from the spring at Ajades some few kilometres north of the ancient city of Samos (modern Pythagoreion) through the mountain north of the city. The tunnel consists of two parts, a north and a south part, obviously cut from both ends. The two tunnel sections meet below the summit of the mountain, which rises to 225 m above mean sea level. The total length of the tunnel is ca. 1040 m divided into 612 m for the north part and 424 m for the south part.[1] The cross-section of the tunnel measures ca. 1.8 x 1.8 m. The floor of the tunnel is almost perfectly flat at ca +55 m. In the tunnel a qanat-like technique has been used for a deep trench with the water pipes at the bottom. Water entered in the north at +51.2 m and left in the south at +47.7 m giving a tilt of 0.36 %. The width of the qanat varies from 0.6 m to 0.8 m. North of the tunnel water was carried, first in a ditch covered with stone slabs and later, before reaching the entrance of the tunnel, a qanat-like technique was used. The construction ends on the south side with an arrangement for distributing water into the city. The total length of the pipe line from the spring to the end in the city is about 3 km using more than 4000 pipes. The inner cross-section of the pipe measures 0.24 to 0.26 m. The capacity has been estimated at 400 cubic meters per 24 hours, equal to ca. 5 litres per second. During construction that probably reached over more than one decade a serious problem arose. When starting the north tunnel it was soon obvious that the material in the mountain was very weak with collapsing walls and roof.[2] This problem was to some extent solved by supporting walls inside the tunnel. But about 270 m into the mountain Eupalinos gave up the straight line and deviated the tunnel to the west to find better material to work in. Further on it was apparently considered safe to turn back towards the original line. Before the rendezvous of the two tunnel sections the north tunnel zigzags, however, in a most challenging way. How Eupalinos here navigated is indeed intriguing. In this paper I argue that the only geometrical component planned to be used by the constructor was the straight line. Because of unforeseen obstacles the tunnelling had to be deviated and for geometrical reasons an equal-legged triangle was used for the navigation. Doing this, different errors were introduced by mistake, which led the tunnelling to a more eastern course. This was later corrected, obviously after a remeasurement, bringing the tunnelling back to the originally chosen straight line. Kienast maintains that it was planned by Eupalinos to secure the south and north tunnels to hit each other by way of a ›tentacle‹ under the summit of the mountain.[3] In my opinion it is only a coincidence that the two tunnels meet each other close to the summit. I argue that the arrangements at the rendezvous were not as claimed by Kienast a method for a secure ›hit‹ by shortest possible tunnelling but instead a shortcut when within hearing distance between the tunnels.[4]

The basic geometrical background information in this paper is taken from H. J. Kienast's description of the tunnel. I have measured angles and distances on Plan 3 a and 3 b of his work and used this for my analysis of the layout of the tunnel. I have used the basic line, introduced by Kienast, and extrapolated through the mountain by him, but renamed it the ›southern line‹. As Kienast, I think Eupalinos used a straight line of poles over the Kastro mountain to determine the starting points of the two tunnels and I have named this line the ›mountain line‹. The line used by Kienast ca. 250 m into the north tunnel is not identical with the ›northern line‹ of this paper. Eupalinos' equal-legged triangle (in theory) has been observed by Kienast and I hold it probable that the triangle was a major component of Eupalinos' navigation. I believe that the north leg of the triangle consists of only two parts.[5] I found a larger deviation in the north tunnel than Kienast and I believe that it was caused by lack of precision in the measurements and not a part of the navigation or a mistake. I use the unit of 20.59 m, determined by Kienast from marks on the walls of the tunnels, to reconstruct the lengths of the tunnel segments and to analyse possible measuring errors. My assumption concerning the setting out of angles is based on simple geometry. This paper deals with Eupalinos' surveying and measurements of the tunnel. By estimating the accuracy obtained by Eupalinos from the geometry of the construction, I have attempted to reconstruct his methodology. My ambition is to approach the work and the constructor through the possibilities available in his time. An empirical approach has been used to understand the layout of the tunnel. The following table identifies key points in the actual cut of the north tunnel used in this paper (figs. 1–2).

The positions of the points A to C are not identical with the wall marks of Eupalinos but they are very close to each other. After E the wall marks representing distances seem to come to an end.

I have used the best available drawing table equipment, rulers and protractors, measuring distances and angles at Plan 3 a of Kienast, and have used Plan 3 a as it were free from errors. The accuracy obtained would be around a quarter of a degree for the angles and 0.25 m for distances. The location of the starting points for the tunnels and the turning points inside the tunnel, used by Eupalinos, can no longer be determined with high precision, which hampers reconstruction and makes it difficult to calculate triangles and traverses more exactly.[7]

I define the different lines as follows:

Southern line: The south tunnel undulates around this line and is the most precise part of the construction from a geometrical view point. This line has been extrapolated and extended theoretically by Kienast through the mountain up to the north entrance. Eupalinos secured the geometry of the line by a vertical shaft situated 20 m into the south tunnel. Obviously this was done to obtain a better connection between this line and the mountain line. It indicates that the possibilities were limited to arrange a sufficient length south of the entrance of the south tunnel for the overlapping of the lines. No other vertical shafts have been found.

Northern line: The north tunnel follows this line for the first 273 m from the entrance. At this point the tunnel turns and the line can no longer be used for practical measurements but remains the base line for the north cut. The tunnel returns to the northern line and follows it from F to G (from 585 m to 601 m). The undulation is not visible in the beginning of the tunnel because of the reinforcements of the walls. Further into the tunnel when the walls can again be observed the undulations seem to be larger compared with those in the south tunnel. The northern line derives from the Kastro mountain line north of the north entrance. Where a straight line could not be established, a parallel line close to the original line was probably used. The southern and the northern lines would in theory be the same line.

Kastro mountain line: This line connects the southern and northern lines. In its south part it is identical with the southern line and in its north part with the northern line. As much of the geometry of the tunnel is dependent on crossing the Kastro mountain this line was probably reworked. Although the Kastro mountain line had a high degree of accuracy, it incorporated a slight curvature, which caused the deviation mentioned below. The curvature would be caused by a minor S-shape in the Kastro mountain line when this line arrived to the northern entrance. It is likely to have occurred on the north slope of the mountain and not at the summit. If it had occurred at the summit it would have been of a magnitude that would have been observed.

These three lines cross each other roughly at the same point in the middle of the north entrance.

Supplementary line: This line guides the tunnel from D to E and was probably, in this part, during the tunnelling, mistaken by the constructor for the northern line.

Deviation: There is a deviation of ca. 0.8 degrees between the northern and the southern lines,[8] and of ca. 1 degree between the northern line and the supplementary line. The northern line as I define it reaches point A 273 m into the north tunnel before turning to the west. According to Kienast the tunnel makes its first turn 235 m into the mountain to the west and soon after to the east.

The deviation was unknown to Eupalinos but he must have been aware of the lower accuracy and risks in transforming the direction by the Kastro mountain line.

Eupalinos had a few important tools at his disposal. 1 Geometry: Eupalinos must have been well acquainted, not only with the geometry that later was collected and developed by Euclid, but also with its practical applications for construction purposes.[9] It is likely that the constructor kept track of his progress in the tunnelling by a full scale model in an open horizontal field in the vicinity of the working site. Kienast suggests that Eupalinos used a drawing table, thus working on a reduced scale.[10] This would have hampered the accuracy considerable.

2 The straight surveying line: The straight surveying line, a line of poles through the landscape, aligned by eye, is still today highly important. The accuracy obtained for a line can, in favourable conditions i.e. flat land with no hindering vegetation, reach a decimetre per kilometre.[11]

It is obvious that Eupalinos employed a surveying line over the Kastro mountain crossing the summit on the western side to avoid the unfavourable central part of the Kastro. As the end of the tunnel had to be situated well within the city walls this also gives a rough position for the exit. The line finally chosen takes a course of 25 degrees NNW (equal to a bearing of 335 degrees).[12]

A more favourable position for the tunnel would have been found about 200 m to the east parallel to the chosen course. The tunnel would have arrived more centrally in the city, the deep trench at the north entrance would have been avoided and it would have been possible for Eupalinos to find a more flat working area at the north entrance. But he seems not to have been able to run the mountain line with sufficient accuracy over the summit of the mountain including the northern part of the ring-wall. This indicates that the walls of the Kastro fortifications were already in place when the survey work for the tunnel started. It is obvious that tunnelling in the dark with only oil-lamps for lighting necessitated a method for keeping a straight course. The tunnel follows the straight line into the mountain in a way that one edge of the tunnel touches the straight line and after this turns to the other side. This makes the walls at the turning points of the tunnel format the line, and it is possible for the surveyor to find the turns by the light from the entrance or a lamp. The undulation will form the tunnel around the straight line. This method avoids keeping a row of rods in working and transport areas. It is most obvious in the south tunnel.[13]

3 Technique for measuring distance: We have no information about how measurements of distances in steep terrain, projected horizontally, were performed. I suppose that a rod of fixed length with an additional water level would do. The accuracy for the horizontally projected distance for the Kastro mountain line from tunnel entrance to tunnel entrance would be around five meters.

In the tunnels are several marks on the walls presumably made for measuring reasons. The marks starting from A excluding words and lines for levelling are in average ca. 4 m apart in both tunnels. But the marks are placed irregularly. Sometimes there are as many as one mark per 2 m and sometimes only one mark per 20 m. On the west wall of both tunnels there is a regular set of marks, with intervals of around 20 m and these marks obviously represent distances. They seem to be arranged in repeated series with ten letters (I K L M N X O P Ω R) each representing a unit length of 20.59.[14] The series starts from both entrances. All marks are today not longer visible or maybe were not placed during the construction but both series can be extrapolated to the entrances giving starting points for the first units. In both tunnels the series come to an end several units before the rendezvous.

It is probable that all parts of the construction work needed measuring marks but today it is almost impossible to connect most of the marks to a specific activity. For cutting the tunnel Eupalinos would have needed the two series of marks showing distance from the tunnel entrances into the mountain and these marks are probably the oldest. Probably several marks are related to a search for possible errors in the measurements. The latter marks would be very difficult to systematise today.

Point A of this paper seems to be close to M and point B to P. Following the logic in the series point C would have been named R, but instead Eupalinos named it S, maybe to emphasise the importance of this turn. The segment C to D is marked according to the series obviously starting with an I which is not visible any longer. The 6.5 unit long segment is not given any mark of this type at its end but instead the O is placed half a unit into the segment D – E.

Distances in horizontal terrain or in the tunnels could have been measured with a chain with or without graduations. As Eupalinos unit is 20.59 m this probably represents the length of the chain. As Eupalinos is using a unit of 20.59 m for distance measurements the chain would have had this length. A chain flat on the floor would give an accuracy of a few centimetres per chain length and for a full segment of the tunnel less than 1 m.[15] The measurements would have followed the straight lines forming the tunnel segments as close as possible preferably with the chain lying on the floor. At the shifts of the chain marks would have been painted on or cut in the floor. The marks on the walls were probably only intended to show the approximate position of the shifts. This would explain why the obvious large spread of distances between the marks on the walls ends up with the length of the tunnel segments in whole or half multiples of units. The marks show both length units and points for shifts of the units.[16] Turning points would have been marked on the floor, precisely at the turning, so that a measuring device could be centrally placed above the point and with space to operate the measurement device around it. No sign of any marks on the floor have up to now been found.

4 Techniques for measuring altitude: Altitude measurements seem to have been made with a chorobates, a primitive leveller with a vessel for water, or forerunners of this instrument. Eupalinos probably measured independently more than once between the south and the north entrances to check his results and to obtain a higher level of accuracy. He would have chosen a flat path around the mountain following a contour line not necessarily on the same level as the entrances.[17] When the two parts meet in the tunnel after three km, the closing error of the levelling is small, only a few decimetres and the accuracy is still better at the entrances.

Eupalinos had probably arranged two stone blocks at each entrance, with flat and levelled top surfaces, aligned with the tunnels, from which the level of the floor of the tunnel could be guided. Possibly a movable wooden pole was used to level it with the surface of the two outer stone blocks. Further into the tunnels lines on the walls seem to have taken over the role of the outer stones. Eupalinos himself seems to have expected a somewhat less precise result as he raised the ceiling of the north tunnel by 2.5 m and lowered the floor of the south tunnel by 0.6 m, when approaching the meeting point. 5 Techniques for measuring angles: The groma and the dioptra are mentioned in classical texts but the groma probably could only handle straight angles and the dioptra was not in use at this time.[18] Nothing resembling a theodolite or a compass was available. Eupalinos probably intended to turn the tunnel at A 15 degrees, at B 7.5 degrees, at C 45 degrees, at D 22.5 degrees, at E 37.5 degrees and at F 37.5 degrees (fig. 1).[19] Eupalinos had to use angles he could deal with. In the construction he was actually not measuring any angles but only setting out angles, which he probably determined from a full-scale model in an open field in the vicinity, as mentioned above. With the available technology and the contemporary level of knowledge of geometry he would probably have used angles easily derived from the straight angle down to a unit not smaller than 1/12 i.e. 7.5 degrees. This gave him angles of 90, 45, 30, 15 and 7.5 degrees, in various combinations. I do not think it was possible to construct any measuring device that could deal with smaller angles than this. With this as a background it is probable that Eupalinos utilized a simple plane table with an aiming device. Combined with the plane table he would have used classical set-squares of somewhat larger size. To increase accuracy he would have used the same set-squares for supplementary and [w:complementary angles|]] in different combinations. It is likely that Eupalinos didn't use tangential-based methods for setting out the directions because of larger difficulties in their use than for using angles and fractions of angles.[20]

Concerning Eupalinos' tools I thus make the following assumptions.

1. The straight line was the main component in Eupalinos' approach to the work and all parts of the tunnels follow straight lines. There are no more turning points in the north tunnel than listed above (A – G).

2. Eupalinos was using measuring units, and I suggest that for angles one unit is equal to 1/12 of a straight angle and for distances one unit is equal to 20.59 m.

3. In the north tunnel, Eupalinos used an equal-legged triangle (in theory) placing the catheti from A to C and from C to D.[21] The angle at D, ¼ of a straight angle (easy to measure today) gives the angles of the equal-legged triangle. This makes the angle at C ½ of a straight angle and the sum of the angles A – B – C ¼ of a straight angle. There is no longer a consistent line of sight today between these points because of reinforcements of the walls and heaps of mud.

The start of the north tunnel achieves a high degree of accuracy; in altitude it is within a few decimetres; the starting point is correct on the lines, also within a few decimetres, but in direction there is a deviation of 0.8 degrees. In length the error is larger–perhaps about 5 m–but this error is less critical. From the situation when starting the north tunnel we can estimate the degree of accuracy possible for Eupalinos.

After 273 meters at A (figs. 3–4), he turned the tunnel 15 degrees (in theory) to the west, perhaps hoping to find better material. He decided the precise turning point after more than 20 m of hesitation. He seems to have changed direction a few degrees on both sides of the turning point before choosing his final western direction. He would, for accuracy, keep the straight northern line as long as possible. After choosing to turn at A he kept this turning point fixed. When leaving A, he would not yet have been able to decide the position of the next turning point because the condition of the rock was unknown for him.

He continued for another 84 m to B where he once more turned the tunnel 7.5 degrees (in theory) to the west and continued 41 m to C. He was by then 40 m (perpendicular distance) out of track of the northern line and 400 m from the north entrance. He had still more than 200 m to go before he reached the south tunnel.[22]

At C he turned the tunnel to the east by 45 degrees (in theory) back towards the northern line. He continued for 133 m, to D, where he turned 22.5 degrees (in theory) westwards to correct his direction south. On his way to D he inadvertently crossed the northern line.

The problem now was that Eupalinos was back on a southbound direction but he was 12 m on the east side of the northern line and 21 m east of the southern line, his goal. He had also increased the deviation to 1.8 degrees compared with the southern line. As he continued almost parallel to the northern line for 34 m he was obviously still not aware of the situation. But at point E something must have happened; he turned the tunnel 37.5 degrees (in theory) to the west to catch up with the northern line which he reached after 20 m at F. Here he again turned 37.5 degrees (in theory) to the east and he was back on the northern line. But he had still 10 m left (perpendicular) to the southern line. The last two turns are close to 39 degrees but, more important, they are equal. That Eupalinos managed to return to the northern line at F and to follow it up to G has to my knowledge not been pointed out before. This was a remarkable achievement.

The use of a comparably sharp turn for correction at E and not a more ›flat‹ angle shows that Eupalinos was eager to come back to the northern line. This suggests that he estimated his position closer to the south tunnel than he actually was. Probably it was at this stage that he attached the ›tentacle‹ to the south tunnel. The choice of a more ›flat‹ angle, for example 15 degrees, would have given more ›tunnel length‹, towards the south. It is unlikely that we will get an explanation of what brought him to make these corrections. Maybe it was a general check-up of measurements and calculations or a re-measurement made as a security step before the rendezvous of the two parts of the tunnel.

A comparison of the segments of the tunnel gives the following relations between Eupalinos´ units and the actual measured distances (fig. 2).[23]

Part of tunnel

Measured distance (m)

Units

(1 unit equivalent to 20.59 m)

Entrance – A

273

A – B

84

4 units (82.4 m)

B – C

41

2 units (41.1 m)

C – D

133

6.5 units (133.8 m)

A – C

124

6 units (123.5 m)

E – F

20

1 unit (20.59 m)

It is of course today difficult to estimate turning points used by Eupalinos but the accuracy would be within a meter. The first turning point at A was chosen by Eupalinos as the best position to turn the line from the construction point of view and not as a full multiple of the length unit counted from the entrance.[24] From D to E the tunnel follows, as Eupalinos would have thought, the correct southward direction, which means that no special care was taken to adapt this segment to one or more full multiples of the unit before he started to correct his position at E.[25]

Eupalinos obtained a remarkable accuracy in several respects, such as in the levelling, in the construction of the more than 400 m long straight line of the south tunnel (shown in the extrapolation of the same line through the mountain arriving almost in the centre of the north entrance), in defining the angles of the equal-legged triangle measured at D to 1/4 of a straight angle, in making the supplementary line almost parallel to the northern line, and in correcting the displacement of the tunnel from E to F, retrieving the northern line at F. But in spite of these outstanding achievements, there were also errors.

Even in modern surveying, errors are common; it is never a ›no-error‹ situation. The question is always how to detect them and control them; they are always there. Even modern engineers are quite discreet about their mistakes.

The angles of the traverse forming the equal-legged triangle, Eupalinos theoretical goal, turn out as follows: 15 + 7.5 + 315 (45) + 22.5 (=360) degrees which bring the traverse back on the same bearing. But the actual measured angles are: 16 degrees at A, 7 degrees at B, 46 degrees at C and 22 degrees at D which adds 1 degree to the deviation. This gives a deviation of 1.8 degrees for the supplementary line, D to E, compared to the southern line. These errors are within the limits of the equipment used and not caused by mistakes. The angle at D between the segment C to D and the supplementary line actually shows a high degree of accuracy.[26]

The two legs of the triangle A – C – D – A should, in theory, be equal in length. Using a rope for checking the critical distances A to C and C to D should give a result with accuracy around one meter for this distance. Even not correcting for the difference in distance between the cathetus A to C or the distance that can be measured A to B plus B to C only adds a minor error. The two catheti are 125 m (A to C) and 133 m (C to D) giving a difference of 8 m which is beyond the measuring inaccuracy. This would have been observed by the constructor. But the undulations between A and C and the turn at B might have confused Eupalinos. The possible use of parallel lines in the tunnel for obtaining straight lines of sight might also have played a role.

Let us recalculate the triangle with the angles at A 22.5 degrees, at C 45 degrees and at D 22.5 degrees and the sides A to C equal to C to D and both 124 m (6 x 20.59 m). In this triangle the corner C of course reaches the northern line.

Another hypothetical calculation can be made. As D and C are better defined as turning points than A we could start the other way round, giving the turning points the same angles but using the distance D to C also for C to A. This gives the position of A far beyond the northern line. These two hypothetical calculations do not add anything to our knowledge of where the error was made, but clearly show that there is an error in this section.[27]

Generally speaking the triangle A – C – D – A seems to show a combination of errors. Some of them could have occurred when turning the tunnel at A, B and C using imprecise angle measurements. Using different distances for the legs in the triangle A – C – D – A is also a source of errors.

A systematic error originates from the deviation of the north tunnel (northern line) of 0.8 degrees compared to the south tunnel (southern line). This deviation causes an error that grows to more than 9 m, perpendicular, at E, 565 m into the tunnel, where Eupalinos realised that something was wrong with his navigation. But he was obviously still not aware of his error in direction because he corrected the tunnel towards the south but only up to the northern line. He only corrected for the difference in lengths of the legs of the triangle A – C – D. Back on the northern line he followed it southwards from F to G. The distance from F to G is too short for making any estimation concerning deviation. At G the north tunnel would be within hearing distance from the south tunnel. If Eupalinos at E (or already at C) had known the position of the ›tentacle‹ or its base, he would have headed for its nearest point and not moved back to the northern line.

When the north tunnel had reached E, Eupalinos would have started cutting the ›tentacle‹ on the south tunnel. If they were started simultaneously in this way (the north tunnel from E and the south tunnel from the beginning of the ›tentacle‹), the ›tentacle‹ would have reached within hearing distance from the north tunnel when this reached G. Eupalinos would then have turned the two parts towards each other. The ›tentacle‹ overcompensates the error in deviation by 50 % which might be seen as a normal safety arrangement.

The south tunnel had, in the meantime, been cut ca. 390 m into the mountain. Here the work, we can assume, had been stopped to await the sorting out of the various difficulties in the north part. The two parts were closing up (fig. 4), and as a precaution Eupalinos attached a ›tentacle‹ to the south tunnel in a direction ca. 30 degrees to the east for ca. 30 m which increased the catching width from the normal tunnel cross-section of 1.8 m to ca. 17 m (perpendicular). Before the rendezvous Eupalinos raised the ceiling of the north tunnel by 2.5 m and lowered the floor of the south tunnel by 0.6 m, giving him a catching height of almost 5 m.

Eupalinos does not seem to have hesitated on which side of the south tunnel he should expect the rendezvous, which suggests that he had also remeasured the mountain line. The ›tentacle‹ would then have been constructed after making the corrections at E. As the work with the south tunnel seems to have proceeded smoothly, the cutting of this tunnel should have continued as far as possible.

But this was not done; the risky navigational situation in the north tunnel had to be resolved first. If we assume the same cutting speed from both sides and a simultaneous start, the head of the north tunnel would have been close to C, which is situated 40 m from the north line, when the work on the south tunnel was stopped. The head of the south tunnel was at this moment situated ca. 15 m south of the summit of the Kastro mountain. Later, when the north tunnel was approaching, the ›tentacle‹ was constructed. In my opinion it was probably a coincidence that the rendezvous took place almost under the summit of the Kastro mountain.[28]

When the tunnels were ca. 12 m apart, both ends were turned towards each other and finally met. I suppose that this last change of direction for both the tunnels was related to the distance a blow with a hammer could be heard through the rock. Since the ›hit‹ between the two tunnels is perpendicular, it is indeed conceivable that it was guided by the sound through the rock. Adding the acoustic based catching width of about 12 m in all directions to the catching width of 17 m of the ›tentacle‹ makes totally about 40 m which is almost 10% of the length of each of the tunnels. The 25 m of acoustic width would have given a sufficient security but obviously Eupalinos did not trust his own measurements at this point.

The only geometrical component planned to be used by Eupalinos was the straight line. Because of unforeseen obstacles the tunnelling had to be deviated and for geometrical reasons an equal-legged triangle was used. Doing this, measuring errors were introduced by mistake. This led the tunnelling to take a more eastern course, which later was corrected after, obviously, remeasurement brought the tunnelling back to the originally chosen straight line.

This line reached the northern entrance with a small deviation of 0.8 degrees which was transformed into the northern line. At an initial stage of the work he did not detect it. The deviation was not a part of the navigation and was not prearranged by Eupalinos but caused by imprecision in the mountain line and the available measurement technology. It cannot have been a part of the construction in order to simplify the rendezvous of the two tunnels. Eupalinos had only one precise tool at his disposal, the straight line. If he had been able to follow his original plan with one straight tunnel from each side he would have known when approaching the rendezvous by measuring the lengths of both tunnels and compared the result with the total length. When hearing the other tunnel he would have changed course if necessary. Hearing distance seems to be around 12 m in this rock material, which would have given him enough security.

Eupalinos was well aware of what level of accuracy he could obtain using the available tools and methodology and he designed the tunnels accordingly. The tunnel is not a miracle but exhibits very skilful craftsmanship. Encountering problems in the rock he was forced to deviate from his original plan. Doing this he incurred errors which he had to solve using the same tools and methodology. Eupalinos was not only a skilful constructor but although he made some measurement errors, he was also a master surveyor. With simple tools and the uttermost determination Eupalinos, and the society of the time supporting him, finalised this remarkable feat of engineering. The pipes transported water to the city for some 1200 years. If the spring were to be restored the tunnel would probably, with some cleaning and maintenance, still work today.

Acknowledgements: I am grateful to the 21st Ephorate of Prehistoric and Classical Antiquities, General Directorate of Antiquities, Athens, for giving me a permit to enter the Eupalinos tunnel, to the staff at the Archaeological Museum of Pythagoreion and at the Eupalinos Tunnel for help and support on my visit, to Mr Rodney Strulo for checking the English text and to Mrs. Bodil Nordström-Karydaki, Cultural Secretary, Swedish Institute, Athens, for assistance in obtaining the permit to enter the tunnel. I also wish to thank Professor Pontus Hellström, Uppsala University, for valuable assistance in the preparation of this article.

Without the excellent publication of the Eupalinos tunnel by H. J. Kienast this study could not have been undertaken.

↑The author has not had access to the background material of the mapping of the tunnel from the German Archaeological Institute. The tunnel has been given an excellent traverse with more than 50 stations by the German Archaeological Institute. Unfortunately the environment of the tunnel has corroded the metal points.

↑I propose that the geometry used derived from complementary and supplementary angles, the sum of the angles of a triangle, the bisection of an angle, equilateral and equal-legged triangles and the Egyptian triangle.

↑Kienast op. cit. (supra n. 1) 150. The unit presented by Kienast is an average of different measurements between 19.45 m to 21.20.

↑A measuring chain or a rope on the flat floor does not need to be corrected for sag. A certain strain must be applied by hand but no influence of temperature would occur because of the regular conditions. A chain is not influenced by moisture. If the handles of the chain were arranged in a way that they could be fixed to a peg, where the centre of the peg defines the end of the chain, a still higher accuracy could be obtained. The peg should fit into a hole drilled in the floor. Using the same chain for all measurements in the tunnels increases the accuracy. Accuracy would be lower if a tape made of fibre or leather were used.

↑Grewe analyses a proposed navigation based on a grid where ›compressed‹ distance units along an extension of the north tunnel are projected perpendicular on the actual cut segments: K. Grewe, Licht am Ende des Tunnels. Planung und Trassierung im Antiken Tunnelbau (Mainz am Rhein 1998) 58–69. This theory is based on the irregular positions of the wall marks and not compatible with possible chain-shifts on the floor of the tunnel. For Eupalinos to arrange the grid in the rock or transfer results from a model of the grid into the rock appears difficult.

↑The angles used for the setting out would have been 1/12 of a straight angle for 7.5 degrees; 2/12 for 15 degrees; 3/12 for 22.5 degrees; 5/12 for 37.5 degrees and 6/12 for 45 degrees. See also above, n. 9.

↑By a tangential-based method I refer to setting out the directions for example by five units ahead and one unit perpendicular to the side, tan 1/5 or 1:5. Kienast argues for the use of the tangential-based method (op. cit. [supra n. 1] 165f. 168 fig. 46a. 169 fig. 46b).

↑It is unlikely that Eupalinos used the shape of the rocks above the north tunnel for navigational purposes with any success as argued by Kienast op. cit. (supra n. 1) 166; Kienast op. cit. (supra n. 21) 72. 79. The slope at C is more than 100 m above the tunnel and at D 150 m. As Eupalinos had no difficulties in cutting through solid rock as shown in the south tunnel he would have avoided all depressions and clefts because of their ability to transport water and instead he would have tried to find solid rock.

↑Kienast suggests that Eupalinos used a length measuring system for the triangular deviation to the west starting from D going backwards towards the north entrance (Kienast op. cit [supra n. 1] 152–154 fig. 41; Kienast op. cit [supra n. 21] 74–78). His result is based on the irregular positions of the wall marks including the word PARADEGMA, between two straight lines; it is not compatible with possible chain-shifts on the floor of the tunnel. A more plausible explanation of the word PARADEGMA has been given by B. Wesenberg (Das Paradeigma des Eupalinos, JdI 122, 2007, 33–49) as related to the construction of the support walls and not to the cutting of the tunnels.

↑Searching for errors in traverses was frequently performed in the pre-computer era by reverse calculation.

↑Kienast (op. cit. [supra n. 1] 146) argues that a main component of the navigation of Eupalinos was that the meeting point of the two tunnels should be under the summit because it gives a possibility to correlate the tunnels with the lines on the slopes. He also maintains (op. cit. [supra n. 1] 140f. 146) that Eupalinos arranged one catching-tunnel and one hitting-tunnel for the “hit” under the summit. The acoustic component is not mentioned.